NAME
Algorithm::CurveFit - Nonlinear Least Squares Fitting
SYNOPSIS
use Algorithm::CurveFit;
# Known form of the formula
my $formula = 'c + a * x^2';
my $variable = 'x';
my @xdata = read_file('xdata'); # The data corresponsing to $variable
my @ydata = read_file('ydata'); # The data on the other axis
my @parameters = (
# Name Guess Accuracy
['a', 0.9, 0.00001], # If an iteration introduces smaller
['c', 20, 0.00005], # changes that the accuracy, end.
);
my $max_iter = 100; # maximum iterations
my $square_residual = Algorithm::CurveFit->curve_fit(
formula => $formula, # may be a Math::Symbolic tree instead
params => \@parameters,
variable => $variable,
xdata => \@xdata,
ydata => \@ydata,
maximum_iterations => $max_iter,
);
use Data::Dumper;
print Dumper \@parameters;
# Prints
# $VAR1 = [
# [
# 'a',
# '0.201366784209602',
# '1e-05'
# ],
# [
# 'c',
# '1.94690440147554',
# '5e-05'
# ]
# ];
#
# Real values of the parameters (as demonstrated by noisy input data):
# a = 0.2
# c = 2
DESCRIPTION
"Algorithm::CurveFit" implements a nonlinear least squares curve fitting
algorithm. That means, it fits a curve of known form (sine-like,
exponential, polynomial of degree n, etc.) to a given set of data
points.
For details about the algorithm and its capabilities and flaws, you're
encouraged to read the MathWorld page referenced below. Note, however,
that it is an iterative algorithm that improves the fit with each
iteration until it converges. The following rule of thumb usually holds
true:
* A good guess improves the probability of convergence and the quality
of the fit.
* Increasing the number of free parameters decreases the quality and
convergence speed.
* Make sure that there are no correlated parameters such as in 'a + b *
e^(c+x)'. (The example can be rewritten as 'a + b * e^c * e^x' in
which 'c' and 'b' are basically equivalent parameters.
The curve fitting algorithm is accessed via the 'curve_fit' subroutine.
It requires the following parameters as 'key => value' pairs:
formula
The formula should be a string that can be parsed by Math::Symbolic.
Alternatively, it can be an existing Math::Symbolic tree. Please refer
to the documentation of that module for the syntax.
Evaluation of the formula for a specific value of the variable
(X-Data) and the parameters (see below) should yield the associated
Y-Data value in case of perfect fit.
variable
The 'variable' is the variable in the formula that will be replaced
with the X-Data points for evaluation. If omitted in the call to
"curve_fit", the name 'x' is default. (Hence 'xdata'.)
params
The parameters are the symbols in the formula whose value is varied by
the algorithm to find the best fit of the curve to the data. There may
be one or more parameters, but please keep in mind that the number of
parameters not only increases processing time, but also decreases the
quality of the fit.
The value of this options should be an anonymous array. This array
should hold one anonymous array for each parameter. That array should
hold (in order) a parameter name, an initial guess, and optionally an
accuracy measure.
Example:
$params = [
['parameter1', 5, 0.00001],
['parameter2', 12, 0.0001 ],
...
];
Then later:
curve_fit(
...
params => $params,
...
);
The accuracy measure means that if the change of parameters from one
iteration to the next is below each accuracy measure for each
parameter, convergence is assumed and the algorithm stops iterating.
In order to prevent looping forever, you are strongly encouraged to
make use of the accuracy measure (see also: maximum_iterations).
The final set of parameters is not returned from the subroutine but
the parameters are modified in-place. That means the original data
structure will hold the best estimate of the parameters.
xdata
This should be an array reference to an array holding the data for the
variable of the function. (Which defaults to 'x'.)
ydata
This should be an array reference to an array holding the function
values corresponding to the x-values in 'xdata'.
maximum_iterations
Optional parameter to make the process stop after a given number of
iterations. Using the accuracy measure and this option together is
encouraged to prevent the algorithm from going into an endless loop in
some cases.
The subroutine returns the sum of square residuals after the final
iteration as a measure for the quality of the fit.
EXPORT
None by default, but you may choose to export "curve_fit" using the
standard Exporter semantics.
SUBROUTINES
This is a list of public subroutines
curve_fit
This subroutine implements the curve fitting as explained in
DESCRIPTION above.
NOTES AND CAVEATS
* When computing the derivative symbolically using "Math::Symbolic", the
formula simplification algorithm can sometimes fail to find the
equivalent of "(x-x_0)/(x-x_0)". Typically, these would be hidden in a
more complex product. The effect is that for "x -> x_0", the
evaluation of the derivative becomes undefined.
Since version 1.05, we fall back to numeric differentiation using
five-point stencil in such cases. This should help with one of the
primary complaints about the reliability of the module.
* This module is NOT fast. For slightly better performance, the formulas
are compiled to Perl code if possible.
SEE ALSO
The algorithm implemented in this module was taken from:
Eric W. Weisstein. "Nonlinear Least Squares Fitting." From MathWorld--A
Wolfram Web Resource.
http://mathworld.wolfram.com/NonlinearLeastSquaresFitting.html
New versions of this module can be found on http://steffen-mueller.net
or CPAN.
This module uses the following modules. It might be a good idea to be
familiar with them. Math::Symbolic, Math::MatrixReal, Test::More
AUTHOR
Steffen Mueller,
COPYRIGHT AND LICENSE
Copyright (C) 2005-2010 by Steffen Mueller
This library is free software; you can redistribute it and/or modify it
under the same terms as Perl itself, either Perl version 5.6 or, at your
option, any later version of Perl 5 you may have available.